2 Key Concepts and SkillsBe able to compute the future value of multiple cash flowsBe able to compute the present value of multiple cash flowsBe able to compute loan paymentsBe able to find the interest rate on a loanUnderstand how interest rates are quotedUnderstand how loans are amortized or paid off6F-2

4 Multiple Cash Flows –Future Value Example 6.1Find the value at year 3 of each cash flow and add them togetherToday (year 0): FV = 7000(1.08)3 = 8,817.98Year 1: FV = 4,000(1.08)2 = 4,665.60Year 2: FV = 4,000(1.08) = 4,320Year 3: value = 4,000Total value in 3 years = 8, , , ,000 = 21,803.58Value at year 4 = 21,803.58(1.08) = 23,547.87The students can read the example in the book. It is also provided here.You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years?Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem, finding the future value at the end for each cash flow and then adding. Point out that you can find the value of a set of cash flows at any point in time, all you have to do is get the value of each cash flow at that point in time and then add them together.I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive, and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.Calculator:Today (year 0 CF): 3 N; 8 I/Y; -7,000 PV; CPT FV = 8,817.98Year 1 CF: 2 N; 8 I/Y; PV; CPT FV = 4,665.60Year 2 CF: 1 N; 8 I/Y; PV; CPT FV = 4,320Year 3 CF: value = 4,000Total value in 3 years = = 21,803.58Value at year 4: 1 N; 8 I/Y; PV; CPT FV = 23,547.87I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.6F-4

7 Multiple Cash Flows – FV Example 3Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?FV = 100(1.08) (1.08)2 = =Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV =Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV =Total FV = =6F-7

8 Multiple Cash Flows – Present Value Example 6.3Find the PV of each cash flows and add themYear 1 CF: 200 / (1.12)1 =Year 2 CF: 400 / (1.12)2 =Year 3 CF: 600 / (1.12)3 =Year 4 CF: 800 / (1.12)4 =Total PV = = 1,432.93The students can read the example in the book.You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one?Point out that the question could also be phrased as “How much is this investment worth?”Calculator:Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV =Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV =Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV =Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV =Total PV = = 1,432.93Remember the sign convention. The negative numbers imply that we would have to pay 1, today to receive the cash flows in the future.6F-8

10 Multiple Cash Flows Using a SpreadsheetYou can use the PV or FV functions in Excel to find the present value or future value of a set of cash flowsSetting the data up is half the battle – if it is set up properly, then you can just copy the formulasClick on the Excel icon for an exampleClick on the tabs at the bottom of the worksheet to move from a future value example to a present value example.Lecture Tip: The present value of a series of cash flows depends heavily on the choice of discount rate. You can easily illustrate this dependence in the spreadsheet on Slide 6.9 by changing the cell that contains the discount rate. A separate worksheet on the slide provides a graph of the relationship between PV and the discount rate.6F-10

12 Multiple Uneven Cash Flows – Using the CalculatorAnother way to use the financial calculator for uneven cash flows is to use the cash flow keysPress CF and enter the cash flows beginning with year 0.You have to press the “Enter” key for each cash flowUse the down arrow key to move to the next cash flowThe “F” is the number of times a given cash flow occurs in consecutive periodsUse the NPV key to compute the present value by entering the interest rate for I, pressing the down arrow and then computeClear the cash flow keys by pressing CF and then CLR WorkThe next example will be worked using the cash flow keys.Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells.Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero.Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.6F-12

13 Decisions, DecisionsYour broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?Use the CF keys to compute the value of the investmentCF; CF0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1NPV; I = 15; CPT NPV = 91.49**No – the broker is charging more than you would be willing to pay.You can also use this as an introduction to NPV by having the students put –100 in for CF0. When they compute the NPV, they will get – You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.6F-13

14 Saving For RetirementYou are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%?Use cash flow keys:CF; CF0 = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5; NPV; I = 12; CPT NPV = 1,084.716F-14

17 Annuities and Perpetuities DefinedAnnuity – finite series of equal payments that occur at regular intervalsIf the first payment occurs at the end of the period, it is called an ordinary annuityIf the first payment occurs at the beginning of the period, it is called an annuity duePerpetuity – infinite series of equal payments6F-17

18 Annuities and Perpetuities – Basic FormulasPerpetuity: PV = C / rAnnuities:Lecture Tip: The annuity factor approach is a short-cut approach in the process of calculating the present value of multiple cash flows and that it is only applicable to a finite series of level cash flows. Financial calculators have reduced the need for annuity factors, but it may still be useful from a conceptual standpoint to show that the PVIFA is just the sum of the PVIFs across the same time period.6F-18

19 Annuities and the CalculatorYou can use the PMT key on the calculator for the equal paymentThe sign convention still holdsOrdinary annuity versus annuity dueYou can switch your calculator between the two types by using the 2nd BGN 2nd Set on the TI BA-II PlusIf you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity dueMost problems are ordinary annuitiesOther calculators also have a key that allows you to switch between Beg/End.6F-19

20 Annuity – Example 6.5You borrow money TODAY so you need to compute the present value.48 N; 1 I/Y; -632 PMT; CPT PV = 23, ($24,000)Formula:The students can read the example in the book.After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?Note that the difference between the answer here and the one in the book is due to the rounding of the Annuity PV factor in the book.6F-20

22 Buying a HouseYou are ready to buy a house, and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000, and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?It might be good to note that the outstanding balance on the loan at any point in time is simply the present value of the remaining payments.6F-22

24 Annuities on the Spreadsheet - ExampleThe present value and future value formulas in a spreadsheet include a place for annuity paymentsClick on the Excel icon to see an example6F-24

25 Quick Quiz – Part IIYou know the payment amount for a loan, and you want to know how much was borrowed. Do you compute a present value or a future value?You want to receive 5,000 per month in retirement. If you can earn 0.75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?CalculatorPMT = 5,000; N = 25*12 = 300; I/Y = .75; CPT PV = 595,808FormulaPV = 5,000[1 – 1 / ] / = 595,8086F-25

26 Finding the PaymentSuppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = % per month). If you take a 4 year loan, what is your monthly payment?20,000 = C[1 – 1 / ] /C =Note if you do not round the monthly rate and actually use 8/12, then the payment will beCalculator:4(12) = 48 N; 20,000 PV; I/Y; CPT PMT =6F-26

27 Finding the Payment on a SpreadsheetAnother TVM formula that can be found in a spreadsheet is the payment formulaPMT(rate,nper,pv,fv)The same sign convention holds as for the PV and FV formulasClick on the Excel icon for an example6F-27

28 Finding the Number of Payments – Example 6.6Start with the equation, and remember your logs.1,000 = 20(1 – 1/1.015t) / .015.75 = 1 – 1 / 1.015t1 / 1.015t = .251 / .25 = 1.015tt = ln(1/.25) / ln(1.015) = months = 7.76 yearsAnd this is only if you don’t charge anything more on the card!You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000.This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students don’t understand how it works, and it is rarely discussed. This is something that students can take away from the class, even if they aren’t finance majors.Calculator:1.5 I/Y1,000 PV-20 PMTCPT N = MONTHS = 7.75 years6F-28

29 Finding the Number of Payments – Another ExampleSuppose you borrow $2,000 at 5%, and you are going to make annual payments of $ How long before you pay off the loan?2,000 = (1 – 1/1.05t) / .05= 1 – 1/1.05t1/1.05t == 1.05tt = ln( ) / ln(1.05) = 3 yearsSign convention matters!!!5 I/Y2,000 PVPMTCPT N = 3 years6F-29

30 Finding the RateSuppose you borrow $10,000 from your parents to buy a car. You agree to pay $ per month for 60 months. What is the monthly interest rate?Sign convention matters!!!60 N10,000 PVPMTCPT I/Y = .75%The next slide talks about how to do this without a financial calculator.6F-30

31 Annuity – Finding the Rate Without a Financial CalculatorTrial and Error ProcessChoose an interest rate and compute the PV of the payments based on this rateCompare the computed PV with the actual loan amountIf the computed PV > loan amount, then the interest rate is too lowIf the computed PV < loan amount, then the interest rate is too highAdjust the rate and repeat the process until the computed PV and the loan amount are equal6F-31

33 Future Values for AnnuitiesSuppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?FV = 2,000( – 1)/.075 = 454,513.04Remember the sign convention!!!40 N7.5 I/Y-2,000 PMTCPT FV = 454,513.04Lecture Tip: It should be emphasized that annuity factor tables (and the annuity factors in the formulas) assumes that the first payment occurs one period from the present, with the final payment at the end of the annuity’s life. If the first payment occurs at the beginning of the period, then FV’s have one additional period for compounding and PV’s have one less period to be discounted. Consequently, you can multiply both the future value and the present value by (1 + r) to account for the change in timing.6F-33

34 Annuity DueYou are saving for a new house, and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?FV = 10,000[(1.083 – 1) / .08](1.08) = 35,061.12Note that the procedure for changing the calculator to an annuity due is similar on other calculators.Calculator2nd BGN 2nd Set (you should see BGN in the display)3 N-10,000 PMT8 I/YCPT FV = 35,061.122nd BGN 2nd Set (be sure to change it back to an ordinary annuity)What if it were an ordinary annuity? FV = 32,464 (so you receive an additional by starting to save today.)6F-34

35 Annuity Due Timeline32,464If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.35,016.126F-35

36 Perpetuity – Example 6.7 Perpetuity formula: PV = C / rCurrent required return:40 = 1 / rr = .025 or 2.5% per quarterDividend for new preferred:100 = C / .025C = 2.50 per quarterThis is a good preview to the valuation issues discussed in future chapters. The price of an investment is just the present value of expected future cash flows.Example statement:Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?6F-36

37 Quick Quiz – Part IVYou want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month?What if the first payment is made today?You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay?Q1: 35(12) = 420 N; 1,000,000 FV; 1 I/Y; CPT PMT =1,000,000 = C ( – 1) / .01C =Q2: Set calculator to annuity due and use the same inputs as above. CPT PMT =1,000,000 = C[( – 1) / .01] ( 1.01)C =Q3: PV = 1.50 / .03 = $506F-37

38 Work the Web ExampleAnother online financial calculator can be found at MoneyChimpClick on the web surfer and work the following exampleChoose calculator and then annuityYou just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years?Money chimp assumes annuity due!!!Payment = $313,497.816F-38

40 Growing Annuity A growing stream of cash flows with a fixed maturity

41 Growing Annuity: ExampleA defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent?6F-41

42 Growing Perpetuity A growing stream of cash flows that lasts foreverLecture Tip: To prepare students for the chapter on stock valuation, it may be helpful to include a discussion of equity as a growing perpetuity.6F-42

43 Growing Perpetuity: ExampleThe expected dividend next year is $1.30, and dividends are expected to grow at 5% forever.If the discount rate is 10%, what is the value of this promised dividend stream?It is important to note to students that in this example the year 1 cash flow was given. If the current dividend were $1.30, then we would need to multiply it by one plus the growth rate to estimate the year 1 cash flow.6F-43

44 Effective Annual Rate (EAR)This is the actual rate paid (or received) after accounting for compounding that occurs during the yearIf you want to compare two alternative investments with different compounding periods, you need to compute the EAR and use that for comparison.6F-44

45 Annual Percentage RateThis is the annual rate that is quoted by lawBy definition APR = period rate times the number of periods per yearConsequently, to get the period rate we rearrange the APR equation:Period rate = APR / number of periods per yearYou should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate6F-45

46 Computing APRs What is the APR if the monthly rate is .5%?.5(12) = 6%What is the APR if the semiannual rate is .5%?.5(2) = 1%What is the monthly rate if the APR is 12% with monthly compounding?12 / 12 = 1%6F-46

47 Things to RememberYou ALWAYS need to make sure that the interest rate and the time period match.If you are looking at annual periods, you need an annual rate.If you are looking at monthly periods, you need a monthly rate.If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly6F-47

48 Computing EARs - ExampleSuppose you can earn 1% per month on $1 invested today.What is the APR? 1(12) = 12%How much are you effectively earning?FV = 1(1.01)12 =Rate = ( – 1) / 1 = = 12.68%Suppose if you put it in another account, you earn 3% per quarter.What is the APR? 3(4) = 12%FV = 1(1.03)4 =Rate = ( – 1) / 1 = = 12.55%Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.6F-48

49 EAR - Formula Remember that the APR is the quoted rate, andm is the number of compounding periods per yearUsing the calculator:The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.6F-49

51 Decisions, Decisions II ContinuedLet’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?First Account:Daily rate = / 365 =FV = 100( )365 =Second Account:Semiannual rate = / 2 = .0265FV = 100(1.0265)2 =You have more money in the first account.It is important to point out that the daily rate is NOT .014, it isFirst Account:365 N; 5.25 / 365 = I/Y; 100 PV; CPT FV =Second Account:2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV =Lecture Tip: Here is a way to drive the point of this section home. Ask how many students have taken out a car loan. Now ask one of them what annual interest rate s/he is paying on the loan. Students will typically quote the loan in terms of the APR. Point out that, since payments are made monthly, the effective rate is actually more than the rate s/he just quoted, and demonstrate the calculation of the EAR.6F-51

52 Computing APRs from EARsIf you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:6F-52

53 APR - ExampleSuppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?On the calculator: 2nd I conv down arrow 12 EFF Enter down arrow 12 C/Y Enter down arrow CPT NOM6F-53

54 Computing Payments with APRsSuppose you want to buy a new computer system and the store is willing to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years, and the interest rate is 16.9% with monthly compounding. What is your monthly payment?Monthly rate = .169 / 12 =Number of months = 2(12) = 243,500 = C[1 – (1 / )24] /C =2(12) = 24 N; 16.9 / 12 = I/Y; 3,500 PV; CPT PMT =See the Lecture Tip in the IM for a discussion of situations that present a mismatch between the payment period and the compounding period.6F-54

56 Present Value with Daily CompoundingYou need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?Daily rate = .055 / 365 =Number of days = 3(365) = 1,095FV = 15,000 / ( )1095 = 12,718.563(365) = 1095 N5.5 / 365 = I/Y15,000 FVCPT PV = -12,718.566F-56

57 Continuous CompoundingSometimes investments or loans are figured based on continuous compoundingEAR = eq – 1The e is a special function on the calculator normally denoted by exExample: What is the effective annual rate of 7% compounded continuously?EAR = e.07 – 1 = or 7.25%6F-57

58 Quick Quiz – Part V What is the definition of an APR?What is the effective annual rate?Which rate should you use to compare alternative investments or loans?Which rate do you need to use in the time value of money calculations?APR = period rate * # of compounding periods per yearEAR is the rate we earn (or pay) after we account for compoundingWe should use the EAR to compare alternativesWe need the period rate, and we have to use the APR to get it6F-58

59 Pure Discount Loans – Example 6.12Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?PV = 10,000 / 1.07 = 9,345.79Remind students that the value of an investment is the present value of expected future cash flows.1 N; 10,000 FV; 7 I/Y; CPT PV = -9,345.796F-59

60 Interest-Only Loan - ExampleConsider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.What would the stream of cash flows be?Years 1 – 4: Interest payments of .07(10,000) = 700Year 5: Interest + principal = 10,700This cash flow stream is similar to the cash flows on corporate bonds, and we will talk about them in greater detail later.6F-60

61 Amortized Loan with Fixed Principal Payment - ExampleConsider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.Click on the Excel icon to see the amortization table6F-61

62 Amortized Loan with Fixed Payment - ExampleEach payment covers the interest expense plus reduces principalConsider a 4 year loan with annual payments. The interest rate is 8%, and the principal amount is $5,000.What is the annual payment?4 N8 I/Y5,000 PVCPT PMT = -1,509.60Click on the Excel icon to see the amortization tableLecture Tip: Consider a $200,000, 30-year loan with monthly payments of $ (7% APR with monthly compounding). You would pay a total of $279,016 in interest over the life of the loan. Suppose instead, you cut the payment in half and pay $ every two weeks (note that this entails paying an extra $ per year because there are 26 two week periods). You will cut your loan term to just under 24 years and save almost $70,000 in interest over the life of the loan. Calculations on TI-BAII plus First: PV = 200,000; N=360; I=7; P/Y=C/Y=12; CPT PMT = (interest = *360 – 200,000) Second: PV = 200,000; PMT = ; I = 7; P/Y = 26; C/Y = 12; CPT N = 614 payments / 26 = years (interest = *614 – 200,000)6F-62

63 Work the Web ExampleThere are web sites available that can easily prepare amortization tablesClick on the web surfer to check out the site and work the following exampleYou have a loan of $25,000 and will repay the loan over 5 years at 8% interest.What is your loan payment?What does the amortization schedule look like?The monthly payment is $6F-63

64 Quick Quiz – Part VIWhat is a pure discount loan? What is a good example of a pure discount loan?What is an interest-only loan? What is a good example of an interest-only loan?What is an amortized loan? What is a good example of an amortized loan?6F-64

65 Ethics IssuesSuppose you are in a hurry to get your income tax refund. If you mail your tax return, you will receive your refund in 3 weeks. If you file the return electronically through a tax service, you can get the estimated refund tomorrow. The service subtracts a $50 fee and pays you the remaining expected refund. The actual refund is then mailed to the preparation service. Assume you expect to get a refund of $978. What is the APR with weekly compounding? What is the EAR? How large does the refund have to be for the APR to be 15%? What is your opinion of this practice?Using a financial calculator to find the APR: PV = 978 – 50 = 928; FV = -978; N = 3 weeks; CPT I/Y = 1.765% per week; APR = (52 weeks per year) = 91.76%!!! Compute the EAR = ( )52 – 1 = %!!!!You would be better off taking a cash advance on your credit card and paying it off when the refund check comes, even if you have the most expensive card available.Refund needed for a 15% APR: PV + 50 = PV(1 + (.15/52))3 PV = $5,761.146F-65

66 Comprehensive ProblemAn investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually?What is the present value of the above if the payments are received at the beginning of each year?If you deposit those payments into an account earning 8%, what will the future value be in 10 years?What will the future value be if you opening the account with $1,000 today, and then make the $100 deposits at the end of each year?Present value problems:End of the year: 10 N; 8 I/Y; 100 PMT; CPT PV =Beginning of the year: PV = $ X 1.08 = $724.69Future value problems:10 N; 8 I/Y; -100 PMT; CPT FV = 1,448.6610N; 8 I/Y; -1,000 PV; -100 PMT; CPT FV = 3,607.586F-66

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